The Torsion Theory Over Pullback And Pullback Rings

Hereditary torsion theories have been developed since the 1950's-1960's and have been extensively studied by Golan, Gabrid, dickson, stenstrom, etc. In this thesis, We discuss the categories of modules constructed by pullback rings and torsion pairs over pullback rings.In the first section, some preliminaries are mentioned. In order to build a good foundation for later parts, we provided some basic concepts and properties of hereditary torsion theories and pullback which can be found in [1][2].In the second section, we extend the pullback to n-pullback, and then study some properties of n-pullback.In the third section, we discuss the relationship amony Gt(M) and AnnM(It), F_t(N_t) and AnnN_t(I₃-t), P_t(M) and M/I_tM, Q_t(N_t) and N_t/I3_tN_t, and then prove the isomorphism of the categories W and M At last we discuss the categories of modules constructed by pullback rings.In the last section.we firstly prove the injective map from R—tors into (R₁ —tors, R₂—tors) by the isomorphism of categories Ι(R) and ε(Ι(R₁),I(R₂)), and then inverstigate the torsion pairs constructed by pullback rings.